\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  4.0 & +  7.00 x_{1} & -10.00 x_{2} & -6.00 x_{3} & -7.00 x_{4} & -1.00 x_{5} & +  1.00 x_{6} & -1.00 x_{7} & +  9.00 x_{8} & + 10.00 x_{9}\\
 x_{11}   &  -16.0 & -10.00 x_{1} & +  5.00 x_{2} & +  5.00 x_{3} & -7.00 x_{4} & +  7.00 x_{5} & + 10.00 x_{6} & +  8.00 x_{7} & +  7.00 x_{8} & -5.00 x_{9}\\
 x_{12}   &  -7.0 & +  3.00 x_{1} & +  9.00 x_{2} & -7.00 x_{3} & +  2.00 x_{4} & -2.00 x_{5} & +  9.00 x_{6} & -1.00 x_{7} & -3.00 x_{8} & +  3.00 x_{9}\\
 x_{13}   &  -3.0 & -2.00 x_{1} & +  9.00 x_{2} &   & +  1.00 x_{4} & -5.00 x_{5} & +  7.00 x_{6} & -4.00 x_{7} & -3.00 x_{8} &   \\
 x_{14}   &  -5.0 & +  5.00 x_{1} & +  6.00 x_{2} & -6.00 x_{3} & -9.00 x_{4} & +  3.00 x_{5} & +  2.00 x_{6} & -8.00 x_{7} & + 10.00 x_{8} & -1.00 x_{9}\\
 x_{15}   &  30.0 & -6.00 x_{1} &   & -1.00 x_{3} & +  9.00 x_{4} &   & +  7.00 x_{6} & -10.00 x_{7} & -8.00 x_{8} & -5.00 x_{9}\\
 x_{16}   &  13.0 & -5.00 x_{1} & +  8.00 x_{2} & -2.00 x_{3} & +  5.00 x_{4} &   & -8.00 x_{6} & -10.00 x_{7} & +  8.00 x_{8} & +  5.00 x_{9}\\
 x_{17}   &  -6.0 & +  8.00 x_{1} & -3.00 x_{2} & +  5.00 x_{3} & -3.00 x_{4} & +  9.00 x_{5} & -1.00 x_{6} & -9.00 x_{7} & +  9.00 x_{8} & -8.00 x_{9}\\
 x_{18}   &  -1.0  &   & +  6.00 x_{2} & -9.00 x_{3} & +  4.00 x_{4} &   & -4.00 x_{6} & +  3.00 x_{7} & +  6.00 x_{8} & -4.00 x_{9}\\
\hline
z    &  0.0 & -5.00 x_{1} & -5.00 x_{2} & -3.00 x_{3} & -5.00 x_{4} &   & -3.00 x_{6} & -5.00 x_{7} &   & -5.00 x_{9}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{9}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,18$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -7.00 y_{10} & + 10.00 y_{11} & -3.00 y_{12} & +  2.00 y_{13} & -5.00 y_{14} & +  6.00 y_{15} & +  5.00 y_{16} & -8.00 y_{17} &   \\
 y_{2}   &  1.0 & + 10.00 y_{10} & -5.00 y_{11} & -9.00 y_{12} & -9.00 y_{13} & -6.00 y_{14} &   & -8.00 y_{16} & +  3.00 y_{17} & -6.00 y_{18}\\
 y_{3}   &  1.0 & +  6.00 y_{10} & -5.00 y_{11} & +  7.00 y_{12} &   & +  6.00 y_{14} & +  1.00 y_{15} & +  2.00 y_{16} & -5.00 y_{17} & +  9.00 y_{18}\\
 y_{4}   &  1.0 & +  7.00 y_{10} & +  7.00 y_{11} & -2.00 y_{12} & -1.00 y_{13} & +  9.00 y_{14} & -9.00 y_{15} & -5.00 y_{16} & +  3.00 y_{17} & -4.00 y_{18}\\
 y_{5}   &  1.0 & +  1.00 y_{10} & -7.00 y_{11} & +  2.00 y_{12} & +  5.00 y_{13} & -3.00 y_{14} &    &   & -9.00 y_{17} &   \\
 y_{6}   &  1.0 & -1.00 y_{10} & -10.00 y_{11} & -9.00 y_{12} & -7.00 y_{13} & -2.00 y_{14} & -7.00 y_{15} & +  8.00 y_{16} & +  1.00 y_{17} & +  4.00 y_{18}\\
 y_{7}   &  1.0 & +  1.00 y_{10} & -8.00 y_{11} & +  1.00 y_{12} & +  4.00 y_{13} & +  8.00 y_{14} & + 10.00 y_{15} & + 10.00 y_{16} & +  9.00 y_{17} & -3.00 y_{18}\\
 y_{8}   &  1.0 & -9.00 y_{10} & -7.00 y_{11} & +  3.00 y_{12} & +  3.00 y_{13} & -10.00 y_{14} & +  8.00 y_{15} & -8.00 y_{16} & -9.00 y_{17} & -6.00 y_{18}\\
 y_{9}   &  1.0 & -10.00 y_{10} & +  5.00 y_{11} & -3.00 y_{12} &   & +  1.00 y_{14} & +  5.00 y_{15} & -5.00 y_{16} & +  8.00 y_{17} & +  4.00 y_{18}\\
\hline
z    &  -0 & -4.00 y_{10} & + 16.00 y_{11} & +  7.00 y_{12} & +  3.00 y_{13} & +  5.00 y_{14} & -30.00 y_{15} & -13.00 y_{16} & +  6.00 y_{17} & +  1.00 y_{18}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 8 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  2.0 & -22.60 y_{10} & -0.60 y_{6} & -18.20 y_{17} & +  6.20 y_{13} & -1.00 y_{8} & +  9.80 y_{15} & +  6.60 y_{16} & +  0.60 y_{2} & -12.60 y_{14}\\
 y_{7}   &  0.0175438596491 & +  9.94 y_{10} & +  0.29 y_{6} & + 16.47 y_{17} & +  2.42 y_{13} & +  0.82 y_{8} & +  5.46 y_{15} & + 13.23 y_{16} & -0.13 y_{2} & + 16.02 y_{14}\\
 y_{3}   &  1.0701754386 & + 26.22 y_{10} & +  0.86 y_{6} & +  3.29 y_{17} & -8.72 y_{13} & +  0.53 y_{8} & +  2.76 y_{15} & -12.29 y_{16} & -1.46 y_{2} & +  4.27 y_{14}\\
 y_{4}   &  1.49122807018 & -5.90 y_{10} & -0.69 y_{6} & -3.34 y_{17} & +  2.39 y_{13} & -0.54 y_{8} & -9.58 y_{15} & +  2.19 y_{16} & +  0.74 y_{2} & +  6.69 y_{14}\\
 y_{12}   &  0.0175438596491 & +  1.39 y_{10} & -0.01 y_{6} & +  0.87 y_{17} & -0.98 y_{13} & +  0.07 y_{8} & -0.64 y_{15} & +  0.03 y_{16} & -0.08 y_{2} & +  0.22 y_{14}\\
 y_{11}   &  0.105263157895 & -1.14 y_{10} & -0.06 y_{6} & -0.76 y_{17} & +  0.13 y_{13} & -0.08 y_{8} & +  0.19 y_{15} & +  0.17 y_{16} & +  0.04 y_{2} & -0.69 y_{14}\\
 y_{5}   &  0.298245614035 & + 11.78 y_{10} & +  0.42 y_{6} & -1.95 y_{17} & +  2.16 y_{13} & +  0.69 y_{8} & -2.60 y_{15} & -1.12 y_{16} & -0.41 y_{2} & +  2.30 y_{14}\\
 y_{18}   &  0.0526315789474 & +  0.53 y_{10} & +  0.07 y_{6} & -0.18 y_{17} & -0.14 y_{13} & -0.04 y_{8} & +  0.79 y_{15} & -1.52 y_{16} & -0.08 y_{2} & -0.75 y_{14}\\
 y_{9}   &  1.68421052632 & -17.77 y_{10} & -0.01 y_{6} & +  0.87 y_{17} & +  3.02 y_{13} & -0.76 y_{8} & + 11.03 y_{15} & -10.31 y_{16} & +  0.09 y_{2} & -6.12 y_{14}\\
\hline
z    &  1.85964912281 & -11.99 y_{10} & -1.02 y_{6} & -0.19 y_{17} & -1.97 y_{13} & -0.81 y_{8} & -30.62 y_{15} & -11.62 y_{16} & -0.03 y_{2} & -5.34 y_{14}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  11.9938596491 & + 22.60 x_{1} & -9.94 x_{7} & -26.22 x_{3} & +  5.90 x_{4} & -1.39 x_{12} & +  1.14 x_{11} & -11.78 x_{5} & -0.53 x_{18} & + 17.77 x_{9}\\
 x_{6}   &  1.01578947368 & +  0.60 x_{1} & -0.29 x_{7} & -0.86 x_{3} & +  0.69 x_{4} & +  0.01 x_{12} & +  0.06 x_{11} & -0.42 x_{5} & -0.07 x_{18} & +  0.01 x_{9}\\
 x_{17}   &  0.189473684211 & + 18.20 x_{1} & -16.47 x_{7} & -3.29 x_{3} & +  3.34 x_{4} & -0.87 x_{12} & +  0.76 x_{11} & +  1.95 x_{5} & +  0.18 x_{18} & -0.87 x_{9}\\
 x_{13}   &  1.96842105263 & -6.20 x_{1} & -2.42 x_{7} & +  8.72 x_{3} & -2.39 x_{4} & +  0.98 x_{12} & -0.13 x_{11} & -2.16 x_{5} & +  0.14 x_{18} & -3.02 x_{9}\\
 x_{8}   &  0.811403508772 & +  1.00 x_{1} & -0.82 x_{7} & -0.53 x_{3} & +  0.54 x_{4} & -0.07 x_{12} & +  0.08 x_{11} & -0.69 x_{5} & +  0.04 x_{18} & +  0.76 x_{9}\\
 x_{15}   &  30.6192982456 & -9.80 x_{1} & -5.46 x_{7} & -2.76 x_{3} & +  9.58 x_{4} & +  0.64 x_{12} & -0.19 x_{11} & +  2.60 x_{5} & -0.79 x_{18} & -11.03 x_{9}\\
 x_{16}   &  11.6245614035 & -6.60 x_{1} & -13.23 x_{7} & + 12.29 x_{3} & -2.19 x_{4} & -0.03 x_{12} & -0.17 x_{11} & +  1.12 x_{5} & +  1.52 x_{18} & + 10.31 x_{9}\\
 x_{2}   &  0.0324561403509 & -0.60 x_{1} & +  0.13 x_{7} & +  1.46 x_{3} & -0.74 x_{4} & +  0.08 x_{12} & -0.04 x_{11} & +  0.41 x_{5} & +  0.08 x_{18} & -0.09 x_{9}\\
 x_{14}   &  5.34035087719 & + 12.60 x_{1} & -16.02 x_{7} & -4.27 x_{3} & -6.69 x_{4} & -0.22 x_{12} & +  0.69 x_{11} & -2.30 x_{5} & +  0.75 x_{18} & +  6.12 x_{9}\\
\hline
z    &  -1.85964912281 & -2.00 x_{1} & -0.02 x_{7} & -1.07 x_{3} & -1.49 x_{4} & -0.02 x_{12} & -0.11 x_{11} & -0.30 x_{5} & -0.05 x_{18} & -1.68 x_{9}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  11.9938596491 & + 22.60 x_{1} & -9.94 x_{7} & -26.22 x_{3} & +  5.90 x_{4} & -1.39 x_{12} & +  1.14 x_{11} & -11.78 x_{5} & -0.53 x_{18} & + 17.77 x_{9}\\
 x_{6}   &  1.01578947368 & +  0.60 x_{1} & -0.29 x_{7} & -0.86 x_{3} & +  0.69 x_{4} & +  0.01 x_{12} & +  0.06 x_{11} & -0.42 x_{5} & -0.07 x_{18} & +  0.01 x_{9}\\
 x_{17}   &  0.189473684211 & + 18.20 x_{1} & -16.47 x_{7} & -3.29 x_{3} & +  3.34 x_{4} & -0.87 x_{12} & +  0.76 x_{11} & +  1.95 x_{5} & +  0.18 x_{18} & -0.87 x_{9}\\
 x_{13}   &  1.96842105263 & -6.20 x_{1} & -2.42 x_{7} & +  8.72 x_{3} & -2.39 x_{4} & +  0.98 x_{12} & -0.13 x_{11} & -2.16 x_{5} & +  0.14 x_{18} & -3.02 x_{9}\\
 x_{8}   &  0.811403508772 & +  1.00 x_{1} & -0.82 x_{7} & -0.53 x_{3} & +  0.54 x_{4} & -0.07 x_{12} & +  0.08 x_{11} & -0.69 x_{5} & +  0.04 x_{18} & +  0.76 x_{9}\\
 x_{15}   &  30.6192982456 & -9.80 x_{1} & -5.46 x_{7} & -2.76 x_{3} & +  9.58 x_{4} & +  0.64 x_{12} & -0.19 x_{11} & +  2.60 x_{5} & -0.79 x_{18} & -11.03 x_{9}\\
 x_{16}   &  11.6245614035 & -6.60 x_{1} & -13.23 x_{7} & + 12.29 x_{3} & -2.19 x_{4} & -0.03 x_{12} & -0.17 x_{11} & +  1.12 x_{5} & +  1.52 x_{18} & + 10.31 x_{9}\\
 x_{2}   &  0.0324561403509 & -0.60 x_{1} & +  0.13 x_{7} & +  1.46 x_{3} & -0.74 x_{4} & +  0.08 x_{12} & -0.04 x_{11} & +  0.41 x_{5} & +  0.08 x_{18} & -0.09 x_{9}\\
 x_{14}   &  5.34035087719 & + 12.60 x_{1} & -16.02 x_{7} & -4.27 x_{3} & -6.69 x_{4} & -0.22 x_{12} & +  0.69 x_{11} & -2.30 x_{5} & +  0.75 x_{18} & +  6.12 x_{9}\\
\hline
z    &  -3.20964912281 & -3.80 x_{1} & -4.77 x_{7} & -7.72 x_{3} & -3.39 x_{4} & -0.42 x_{12} & -0.01 x_{11} & -0.80 x_{5} & -0.20 x_{18} & -4.58 x_{9}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  11.9938596491 & + 22.60 x_{1} & -9.94 x_{7} & -26.22 x_{3} & +  5.90 x_{4} & -1.39 x_{12} & +  1.14 x_{11} & -11.78 x_{5} & -0.53 x_{18} & + 17.77 x_{9}\\
 x_{6}   &  1.01578947368 & +  0.60 x_{1} & -0.29 x_{7} & -0.86 x_{3} & +  0.69 x_{4} & +  0.01 x_{12} & +  0.06 x_{11} & -0.42 x_{5} & -0.07 x_{18} & +  0.01 x_{9}\\
 x_{17}   &  0.189473684211 & + 18.20 x_{1} & -16.47 x_{7} & -3.29 x_{3} & +  3.34 x_{4} & -0.87 x_{12} & +  0.76 x_{11} & +  1.95 x_{5} & +  0.18 x_{18} & -0.87 x_{9}\\
 x_{13}   &  1.96842105263 & -6.20 x_{1} & -2.42 x_{7} & +  8.72 x_{3} & -2.39 x_{4} & +  0.98 x_{12} & -0.13 x_{11} & -2.16 x_{5} & +  0.14 x_{18} & -3.02 x_{9}\\
 x_{8}   &  0.811403508772 & +  1.00 x_{1} & -0.82 x_{7} & -0.53 x_{3} & +  0.54 x_{4} & -0.07 x_{12} & +  0.08 x_{11} & -0.69 x_{5} & +  0.04 x_{18} & +  0.76 x_{9}\\
 x_{15}   &  30.6192982456 & -9.80 x_{1} & -5.46 x_{7} & -2.76 x_{3} & +  9.58 x_{4} & +  0.64 x_{12} & -0.19 x_{11} & +  2.60 x_{5} & -0.79 x_{18} & -11.03 x_{9}\\
 x_{16}   &  11.6245614035 & -6.60 x_{1} & -13.23 x_{7} & + 12.29 x_{3} & -2.19 x_{4} & -0.03 x_{12} & -0.17 x_{11} & +  1.12 x_{5} & +  1.52 x_{18} & + 10.31 x_{9}\\
 x_{2}   &  0.0324561403509 & -0.60 x_{1} & +  0.13 x_{7} & +  1.46 x_{3} & -0.74 x_{4} & +  0.08 x_{12} & -0.04 x_{11} & +  0.41 x_{5} & +  0.08 x_{18} & -0.09 x_{9}\\
 x_{14}   &  5.34035087719 & + 12.60 x_{1} & -16.02 x_{7} & -4.27 x_{3} & -6.69 x_{4} & -0.22 x_{12} & +  0.69 x_{11} & -2.30 x_{5} & +  0.75 x_{18} & +  6.12 x_{9}\\
\hline
z    &  -3.20964912281 & -3.80 x_{1} & -4.77 x_{7} & -7.72 x_{3} & -3.39 x_{4} & -0.42 x_{12} & -0.01 x_{11} & -0.80 x_{5} & -0.20 x_{18} & -4.58 x_{9}\\
\end{array}\]
Final Dictionary
Solution:  -3.20964912281
Num Pivots:  1
\end{document}
